A consistent example of a hereditarily 𝔠-Lindelöf first countable space of size >𝔠

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

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The convergence-theoretic approach to weakly first countable spaces and symmetrizable spaces

Mathematica Slovaca ◽

10.1515/ms-2017-0213 ◽

2019 ◽

Vol 69 (1) ◽

pp. 185-198

Author(s):

Fadoua Chigr ◽

Frédéric Mynard

Keyword(s):

Space Structures ◽

Theoretic Approach ◽

Hausdorff Spaces ◽

Sequential Space ◽

Countable Space ◽

The Stability ◽

First Countable Space ◽

Algebraic Proofs ◽

Non Hausdorff Spaces

AbstractThis article fits in the context of the approach to topological problems in terms of the underlying convergence space structures, and serves as yet another illustration of the power of the method. More specifically, we spell out convergence-theoretic characterizations of the notions of weak base, weakly first-countable space, semi-metrizable space, and symmetrizable spaces. With the help of the already established similar characterizations of the notions of Frchet-Ursyohn, sequential, and accessibility spaces, we give a simple algebraic proof of a classical result regarding when a symmetrizable (respectively, weakly first-countable, respectively sequential) space is semi-metrizable (respectively first-countable, respectively Fréchet) that clarifies the situation for non-Hausdorff spaces. Using additionally known results on the commutation of the topologizer with product, we obtain simple algebraic proofs of various results of Y. Tanaka on the stability under product of symmetrizability and weak first-countability, and we obtain the same way a new characterization of spaces whose product with every metrizable topology is weakly first-countable, respectively symmetrizable.

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What is a first countable space?

Topology and its Applications ◽

10.1016/j.topol.2006.03.003 ◽

2006 ◽

Vol 153 (18) ◽

pp. 3420-3429 ◽

Cited By ~ 1

Author(s):

Gonçalo Gutierres

Keyword(s):

Countable Space ◽

First Countable Space

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Different versions of a first countable space without choice

Topology and its Applications ◽

10.1016/j.topol.2009.03.025 ◽

2009 ◽

Vol 156 (12) ◽

pp. 2000-2004

Author(s):

Kyriakos Keremedis ◽

Eleftherios Tachtsis

Keyword(s):

Countable Space ◽

First Countable Space

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A better framework for first countable spaces

Applied General Topology ◽

10.4995/agt.2003.2034 ◽

2003 ◽

Vol 4 (2) ◽

pp. 289

Author(s):

Gerhard Preuss

Keyword(s):

Topological Space ◽

Uniform Space ◽

Topological Spaces ◽

Convergence Space ◽

Convergence Spaces ◽

Natural Function ◽

Topological Construct ◽

Countable Space ◽

Filter Space ◽

First Countable Space

<p>In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.</p>

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Separating Points and Coloring Principles

Canadian Mathematical Bulletin ◽

10.4153/cmb-1984-061-3 ◽

1984 ◽

Vol 27 (4) ◽

pp. 398-404

Author(s):

W. Stephen Watson

Keyword(s):

Binary Tree ◽

Weak Version ◽

Countable Space ◽

First Countable Space ◽

Discrete Family

AbstractIn the mid 1970's, Shelah formulated a weak version of ◊. This axiom Φ is a prediction principle for colorings of the binary tree of height ω1. Shelah and Devlin showed that Φ is equivalent to 2ℵ0 < 2ℵ1.In this paper, we formulate Φp, a "Φ for partial colorings", show that both ◊* and Fleissner's “◊ for stationary systems” imply Φp, that ◊ does not imply Φp and that Φp does not imply CH.We show that Φp implies that, in a normal first countable space, a discrete family of points of cardinality ℵ1 is separated.

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Compact complement topologies and k-spaces

Filomat ◽

10.2298/fil1907061k ◽

2019 ◽

Vol 33 (7) ◽

pp. 2061-2071

Author(s):

K. Keremedis ◽

C. Özel ◽

A. Piękosz ◽

Shumrani Al ◽

E. Wajch

Keyword(s):

Hausdorff Space ◽

Multiple Choice ◽

Metrizable Space ◽

Paper Properties ◽

Countable Space ◽

Sorgenfrey Line ◽

First Countable Space ◽

Infinite Set

Let (X,?) be a Hausdorff space, where X is an infinite set. The compact complement topology ?* on X is defined by: ?* = {0}?{X\M:M is compact in (X,?)}. In this paper, properties of the space (X,?*) are studied in ZF and applied to a characterization of k-spaces, to the Sorgenfrey line, to some statements independent of ZF, as well as to partial topologies that are among Delfs-Knebusch generalized topologies. Between other results, it is proved that the axiom of countable multiple choice (CMC) is equivalent with each of the following two sentences: (i) every Hausdorff first-countable space is a k-space, (ii) every metrizable space is a k-space. A ZF-example of a countable metrizable space whose compact complement topology is not first-countable is given.

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Cardinal estimates involving the weak Lindelöf game

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01141-0 ◽

2021 ◽

Vol 116 (1) ◽

Author(s):

Leandro Aurichi ◽

Angelo Bella ◽

Santi Spadaro

Keyword(s):

Compact Space ◽

Winning Strategy ◽

Partial Answer ◽

Large Cardinality ◽

Countable Space ◽

First Countable Space

AbstractWe show that if X is a first-countable Urysohn space where player II has a winning strategy in the game $$G^{\omega _1}_1({\mathcal {O}}, {\mathcal {O}}_D)$$ G 1 ω 1 ( O , O D ) (the weak Lindelöf game of length $$\omega _1$$ ω 1 ) then X has cardinality at most continuum. This may be considered a partial answer to an old question of Bell, Ginsburg and Woods. It is also the best result of this kind since there are Hausdorff first-countable spaces of arbitrarily large cardinality where player II has a winning strategy even in the weak Lindelöf game of countable length. We also tackle the problem of finding a bound on the cardinality of a first-countable space where player II has a winning strategy in the game $$G^{\omega _1}_{fin}({\mathcal {O}}, {\mathcal {O}}_D)$$ G fin ω 1 ( O , O D ) , providing some partial answers to it. We finish by constructing an example of a compact space where player II does not have a winning strategy in the weak Lindelöf game of length $$\omega _1$$ ω 1 .

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The partially pre-ordered set of compactifications of Cp(X, Y)

Topological Algebra and its Applications ◽

10.1515/taa-2015-0002 ◽

2015 ◽

Vol 3 (1) ◽

Author(s):

A. Dorantes-Aldama ◽

R. Rojas-Hernández ◽

Á. Tamariz-Mascarúa

Keyword(s):

Ordered Set ◽

Continuous Functions ◽

Space Of Continuous Functions ◽

Countable Space ◽

Generalized Ordered Space ◽

Ordered Space ◽

Countable Character ◽

Corson Compact ◽

First Countable Space ◽

Isolated Point

AbstractIn the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

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Notes on Fréchet spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299226592 ◽

1999 ◽

Vol 22 (3) ◽

pp. 659-665 ◽

Cited By ~ 2

Author(s):

Woo Chorl Hong

Keyword(s):

Topological Space ◽

Sufficient Conditions ◽

Continuous Functions ◽

Sufficient Condition ◽

Fréchet Spaces ◽

Sequential Space ◽

Countable Space ◽

Sequential Closure ◽

Simple Expansion ◽

First Countable Space

First, we introduce sequential convergence structures and characterize Fréchet spaces and continuous functions in Fréchet spaces using these structures. Second, we give sufficient conditions for the expansion of a topological space by the sequential closure operator to be a Fréchet space and also a sufficient condition for a simple expansion of a topological space to be Fréchet. Finally, we study on a sufficient condition that a sequential space be Fréchet, a weakly first countable space be first countable, and a symmetrizable space be semi-metrizable.

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